England rode on Ben Stokes's brilliant hundred and some incisive bowling by Mark Wood and Adil Rashid, who returned with four-wicket hauls each at Edgbaston here.
Opting to bowl, Wood (4/33) and Rashid (4/41) returned with superb four-wicket hauls as Australia suffered a late batting collapse to post 277-9.
Aaron Finch (68) and Steve Smith (56) scored twin half centuries at the top to lay the foundation but Australia lost five wickets within 15 runs to lose the momentum. It was Travis Head's unbeaten 71-run innings which helped them manage a respectable score.
Chasing 278 to win, England were reduced to 35-3 in the 6th over but Eoin Morgan (87) and Ben Stokes (102) resurrected England's run chase with a strokeful 159-run partnership in 26.1 overs for the fourth wicket here.
After Morgan's dismissal, Stokes and Jos Butler took England close to victory, reaching 240-4 in 40.2 overs, before another rain interruption brought an early end to the game as England were declared winners by 40 runs in Duckworth-Lewis method.
Defending the total, Australian pacer Mitchell Starc dealt a blow with his second ball itself, dismissing opener Jason Roy (4) in the first over.
Seamer Josh Hazlewood then dismissed Alex Hales (0), before getting rid of Joe Root (15) early to leave England tottering at 35-3 in the 6th over when heavens opened up and play was stopped for around 40 minutes.
After resumption, Morgan and Stokes brought up the fifty for England in 7.4 overs. The duo batted with purpose and piled up the 100 in 14.4 overs.
The two batsmen shared their 100-run partnership in 93 balls, before taking England across the 150-mark in the 24th over.
In the 32nd over, however, a terrible mix-up cost Morgan his wicket as England slipped to 194-4.
Stokes and Butler (29) then guided England to 240-4 in 40.2 overs when rain once again returned to interrupt the play. The Match was then abandoned and England were declared winners by 40 runs on Duckworth-Lewis method.
Earlier, in their must-win encounter, World Champions Australia lost opener David Warner (21) in the eighth over when he edged a Wood delivery and was caught by wicket-keeper Jos Butler.
Finch and Smith then dug their heels and ensured Australia don't suffer any more hiccups and brought up the first 50 runs in 8.5 overs.
The duo rotated the strike well and picked up odd boundaries to guide Australia across the 100-mark in 18 overs.
Meanwhile, Finch completed his 50 off 47 balls in the 17th over with the help of seven fours. But he became a victim of Stokes in the 23rd over when a miscued shot was smartly taken by Eoin Morgan, rushing from mid-off.
At 136-2, Moises Henriques joined Smith and the duo brought up the 150 in 25.3 overs. But the former couldn't stay longer as his miscued slog shot was holed out at mid-on by Plunkett off Rashid.
In the 31st over, Smith sent Plunkett for a boundary to reach his fifty off 68 balls with the help of five fours. One over later, Plunkett caught the Australian captain at mid-off after he failed to negotiate a Wood delivery.
Head and Glenn Maxwell (20) then shared 58 runs off 62 balls to take Australia to 239.
However, Australia suffered a collapse after that losing Maxwell, Matthew Wade (2), Mitchell Starc (0), Pat Cummins (4) and Adam Zampa (0) within 15 runs as they failed to accelerate in the end overs.
Head hit some lusty blows towards the end to prop up Australia.